When I read your question I thought of two things:
Stress can definitely intensify underlying or pre existing conditions. As stated in your wikipedia link above, "the scratch refex is a response to activation of sensory neurons whose peripheral terminals are located on the surface of the body." If your pup has an underlying condition, whether noticeable or not (anywhere from an insect bite but not limited to dry skin, allergic dermatitis, yeast, ringworm, or alopecia); the stress response in his body will stimulate neurofibrils aka nerve fibers already affected by skin condition, thus causing an itch, or make it worse.
This can also happen if there IS NOT a previous condition. Stress and anxiety ultimately affects your pup's nervous system. When your dog is experiencing anxiety they produce stress hormones like dopamine and adrenaline. Those chemicals send messages through the nervous system via nerve cells. Nerve cells communicate with muscular cells and muscular cells communicate with skin cells, thus causing the scratch reflex you speak of. Some of my answer about stress hormones in dogs were found here: http://www.doglistener.tv/2015/05/stress-in-dogs-what-we-cant-see/
Another idea that I have is allergies due to changes: There are a number of things in the new environment that could be causing an allergic reaction, ie the random scratches. Your dog may be exposed to allergins whether it's sneaking or sharing the other dogs food or treats, shampoo, sprays, topical medications, or detergent used on bedding. There could also be molds and mildews in the home, fleas or mites left from previous tenant or the other dog. The environment outside of the home may possibly be different than your previous location with more exposure to grass, pollen, ragweeds etc (Not sure if you moved to a different neighborhood, town or state). It may be a mild allergy also explaining why it's not constant scratching. Here's a link for addt'l allergens and what to do if your dog is allergic:
Most allergic reactions aren't limited to a certain spot (if he's only having a scratch reflex on his rear, for example), BUT having flea dermatitis or the beginnings of mange would start in a certain area of his body.
Although I'm sure the first part of my answer, stress response, is what you're dog is dealing with, it might be of interest to explore the other possibilities. Good & interesting question!